Olympiad problems

Durer Math Competition CD 1st Round - geometry, 2009.D4

If all vertices of a triangle on the square grid are grid points, then the triangle is called a [i]lattice[/i] triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area $1$ square unit?

1997 IMO Shortlist, 12

Tags: algebra , polynomial , modular arithmetic , congruence

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

2004 China Western Mathematical Olympiad, 2

Tags: incenter , exterior angle , geometry

Let $ABCD$ be a convex quadrilateral, $I_1$ and $I_2$ be the incenters of triangles $ABC$ and $DBC$ respectively. The line $I_1I_2$ intersects the lines $AB$ and $DC$ at points $E$ and $F$ respectively. Given that $AB$ and $CD$ intersect in $P$, and $PE=PF$, prove that the points $A$, $B$, $C$, $D$ lie on a circle.

1997 Federal Competition For Advanced Students, Part 2, 2

Tags: modular arithmetic , number theory

A positive integer $K$ is given. Define the sequence $(a_n)$ by $a_1 = 1$ and $a_n$ is the $n$-th positive integer greater than $a_{n-1}$ which is congruent to $n$ modulo $K$. [b](a)[/b] Find an explicit formula for $a_n$. [b](b)[/b] What is the result if $K = 2$?

2018 Romania Team Selection Tests, 1

Tags: angle chasing , projective geometry , similar triangles , geometry

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2012 AMC 12/AHSME, 10

Tags: geometry , trigonometry

A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? $ \textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10} $

1971 IMO Shortlist, 12

Tags: midpoint , geometry , congruent triangles , collinear

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

1999 Iran MO (2nd round), 2

Tags: symmetry , vector , geometry

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

2013 Saudi Arabia BMO TST, 6

Tags: midpoint , geometry , incenter , equal segments

Let $ABC$ be a triangle with incenter $I,$ and let $D,E,F$ be the midpoints of sides $BC, CA, AB$, respectively. Lines $BI$ and $DE$ meet at $P $ and lines $CI$ and $DF$ meet at $Q$. Line $PQ$ meets sides $AB$ and $AC$ at $T$ and $S$, respectively. Prove that $AS = AT$

2011 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: combinatorics , round table

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

1980 IMO, 1

Tags: algebra , arithmetic sequence , arithmetic progression , sequence

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

2016 Harvard-MIT Mathematics Tournament, 10

Tags:

Let $a,b$ and $c$ be real numbers such that \begin{align*} a^2+ab+b^2&=9 \\ b^2+bc+c^2&=52 \\ c^2+ca+a^2&=49. \end{align*} Compute the value of $\dfrac{49b^2+39bc+9c^2}{a^2}$.

2011 Laurențiu Duican, 3

Tags: function , real analysis , continuity , fractional part

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

1990 IMO Shortlist, 5

Tags: geometry , incenter , vector , euler , trigonometry , orthocenter

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

2004 Flanders Math Olympiad, 1

Tags: perimeter , incenter , ratio , circumcircle , geometry

[u][b]The author of this posting is : Peter VDD[/b][/u] ____________________________________________________________________ most of us didn't really expect to get this, but here it goes (flanders mathematical olympiad 2004, today) triangle with sides 501m, 668m, 835m how many lines can be draws so that the line halves both area and circumference?

2005 iTest, 18

Tags: geometry , triangle inequality

If the four sides of a quadrilateral are $2, 3, 6$, and $x$, find the sum of all possible integral values for $x$.

2022 APMO, 4

Tags: combinatorics

Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled $1$ to $n$. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say $i$, to either any empty box or the box containing marble $i+1$. Cathy wins if at any point there is a box containing only marble $n$. Determine all pairs of integers $(n,k)$ such that Cathy can win this game.

2017 Saudi Arabia JBMO TST, 5

Tags: algebra , inequality

Let $a,b,c>0$ and $a+b+c=6$ . Prove that $$ \frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16} \ge \frac{1}{8}.$$

2020 Purple Comet Problems, 10

Tags: number theory

Given that $a, b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, the minimum possible positive value of $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$, is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2008 Princeton University Math Competition, B1

Tags: number theory

What is the remainder, in base $10$, when $24_7 + 364_7 + 43_7 + 12_7 + 3_7 + 1_7$ is divided by $6$?

2025 Malaysian APMO Camp Selection Test, 5

Tags: combinatorics , geometry , combinatorial geometry

Fix a positive integer $n\ge 2$. For any cyclic $2n$-gon $P_1 P_2\cdots P_{2n}$ in this order, define its score as the maximal possible value of $$\angle P_iXP_{i+1} + \angle P_{i+n}XP_{i+n+1}$$ across all $1\le i\le n$ (indices modulo $n$), and over all points $X$ inside the $2n$-gon including its boundary. Prove that there exist a real number $r$ such that a cyclic $2n$-gon is regular if and only if it has score $r$. [i]Proposed by Wong Jer Ren[/i]

2014 Czech-Polish-Slovak Junior Match, 5

Tags: polygon , square , sum , combinatorial geometry , combinatorics , angle

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

2015 BMT Spring, P2

Tags: polynomial , algebra

Let $f(x)$ be a nonconstant monic polynomial of degree $n$ with rational coefficents that is irreducible, meaning it cannot be factored into two nonconstant rational polynomials. Find and prove a formula for the number of monic complex polynomials that divide $f$.

2021 Irish Math Olympiad, 6

Tags: geometry , equilateral , algebra , arithmetic progression

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

2022 Indonesia TST, C

Tags: counting , combinatorics

Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.